12. 800: Fluid Dynamics of the Atmosphere and Ocean

Chapter 1The continuum hypothesis and kinematics

1.1 Introduction and syllabus.

This course is an introduction to fluid mechanics with special attention paid to conceptsand applications that are important in oceanography and meteorology. Indeed, bothmeteorology and oceanography are notable for the fact that the explanation of fundamentalphenomena requires a deep understanding of fluid mechanics. Phenomena like the GulfStream, coastal upwelling, the polar stratospheric vortex, the Jet Stream can all beapproached as fundamental problems in fluid mechanics.

It is, as I hope to persuade you in this course, a beautiful subject dealing withcompellingly beautiful physical phenomena. Even on the smallest scales accessible ineveryday life there are lovely things to see and think about. Turn on the faucet in your sinkand watch the roughly circular wave that surrounds the location where the water hits the sinkand speeds outward, watch and wonder as the surf nearly always approach the beach head-on, see wave patterns in the clouds. Or, watch the smooth flow of water moving over a weirand wonder why it is so beautifully smooth. Why does each little pebble on the beach havea V-shaped pattern scoured in the sand behind it? Why is there weather instead of just tidalrepetition? Why does a hurricane resemble a spiral galaxy? How can a tsunami move sofast? Why are there tornados, Dorothy, in Kansas but rarely in California? These are only afew questions that lead us to examine the dynamics of fluids. My own belief is thatequipped with an understanding and sensitivity to the dynamics of fluids life can never beboring; there is always something fascinating to see and think about all around us.

A tentative syllabus (including more than we can probably get to) for the course mightbe as follows;1) Formulation:a) Continuum hypothesisb) Kinematics, Eulerian and Lagrangian descriptions. (rates of change)c) Definitions of streamlines, tubes, trajectories.

Chapter 1 2

2) a) Mass conservation. (For fixed control volume). Careful definition of incompressibility

and distinction with energy equation for ρ = constant.

b) Newton’s lawc) Cartesian tensors, transformations, quotient law for vectors and tensors, diagonalization.d) Tensor character of surface forces (derivation)Σ i = σ ijnje) Momentum equation for a continuum.f) Conservation of angular momentum, symmetry of stress tensor. Trace of stress tensor,mechanical definition of pressure in terms of trace.g) Stress, rate-of- strain tensor, isotropic fluids. Relation between mechanical andthermodynamic pressure. Fluid viscosity, kinematic viscosity. Navier-Stokes equations.f) Navier-Stokes equations in a rotating frame, centripetal and Coriolis accelerations.g) Boundary conditions, kinematical and dynamical—surface tension.3) Examples for a fluid of constant density.a) Ekman layer, Ekman spiral, dissipation in boundary layer, spin-down time, Ekmanpumping. Ekman layer with applied wind-stress. Qualitative discussion of Prandtl boundarylayer.b) The impulsively started flat plate. Diffusion of momentum.4) Thermodynamics.a)State equation, perfect gas, Internal energy.b) First law of thermodynamics. Removal of mechanical contribution and equation forinternal energy. Dissipation function.c) Dissipation due to difference between mechanical and thermodynamic pressure.d) entropy, enthalpy, temperature equation for a perfect gas. Potential temperature. Liquids.e) temperature change in water for the impulsively started flat plate due to friction.5) Fundamental theorems-Vorticitya) Vorticity, definition, vortex lines, tubes, non-divergence. Vortex tube strength.b) Circulation and relation to vorticity.c) Kelvin’s theorem, interpretation. Friction, baroclinicity. Effect of rotation, Induction ofrelative vorticity on the sphere. Rossby waves as an example.d) Bathtub vortex.e) Vortex lines frozen in fluid. Helmholtz theorem.f) The vorticity equation, interpretation.g) Enstrophy.h) Ertel’s theorem of potential vorticity. Relation to Kelvin’s theorem. PV in ahomogeneous layer of fluid. The status function.

Chapter 1 3

i) Thermal wind. Taylor-Proudman theorem. Geostrophy in atmospheric and oceanicdynamics. Consequences.j) Simple scaling arguments and heuristic derivation of quasi-geostrophic PV equation.

6. Fundamental Theorems. Bernoulli statements

a) Energy equation for time dependent dissipative motion.b) Bernoulli theorem for steady inviscid flow. The Bernoulli function B.c) Crocco’s theorem ( relation of grad B to entropy gradients and vorticity.)d) Bernoulli’s theorem for a barotropic fluid.e) Shallow water theory and the pv equation and Bernoulli equation. Relation betweengradB and potential vorticity and stream function.f) Potential flow and the Bernoulli theorem.APPLICATIONS AND EXAMPLESg) Surface gravity waves as an example of (f)h) Waves, trajectories, streamlines.i) Two streams, internal waves with currents. Kelvin-Helmholtz instability. Richardsonnumber Miles-Howard theorem. Effect of surface tension. Physical interpretation ofinstability.j) Brief discussion of baroclinic instability. Wedge of instability, estimate of growth rates.

POSSIBLE SPECIAL TOPICSi) Potential flow past a cylinder. Circulation, Lift (why airplanes fly).ii) Vortex motion in 2-d. Point vortices. Hill vortex. Poincare theorems for continuousdistributions of vorticity. Modons. Effect of beta. Stern-Flierl theorem.iii)Low Reynolds number flow. Down-welling in the mantle.

There are quite a number of excellent book on fluid mechanics. Styles vary and somepeople prefer one to another. There is no required text for the course and if you choose atext it should be clear from what we discuss in class what the appropriate readings might bethat are helpful and I will generally leave reading assignments to you.

I list here a few books that I believe are useful.1) Kundu, P.K. Fluid Mechanics. Academic Press 1990. pp 638.2) Batchelor, G.K. Fluid Dynamics. Cambridge Univ. Press. 1967. pp 615 (This is

one of my favorites).

Chapter 1 4

3) Aris, R. Vectors, Tensors and the basic equations of fluid mechanics. 1962.Prentice Hall. pp 286.

4) Cushman-Roisin, B. Introduction to Geophysical Fluid Dynamics. 1994. Prentice-Hall. pp320

5) Gill, A.E., Atmospheric-Ocean Dynamics. 1982. Academic Press pp 662.6) Pedlosky, J. Geophysical Fluid Dynamics 1987. Springer Verlag pp 710.

1.2 What is a fluid?Naturally, one of the first issues to deal with is getting a clear understanding of what

we mean by a fluid. It is surprisingly difficult to be rigorous about this but fortunately thematerials we are mostly interested in, air and water, clearly fall under the definition of a fluidwe will shortly give. However, in general it is sometimes unclear whether a material isalways either a fluid or a solid.

A perfectly elastic solid is defined and recognized to be a material which suffers aproportionally small and reversible displacement when acted upon by a small force. If, forexample, we take a cubic lump of steel and apply a force tangent to the surface of the cubeas shown in the figure we observe a small displacement of the material.

F

Figure 1.2.1 A small elastic cube subject to a force F suffers a small displacement

with an angle θ.

If the solid is elastic the removal of the force will lead to a restoration of the originalsituation. For a fluid the same small tangential force will lead to an unbounded

displacement. Instead, it is the rate of increase of θ that will be proportional to the force (or

more precisely, the force per unit area to which it is applied. Hence, for a fluid suchtangential forces lead to continuous deformations whose rate is proportional to the force.Thus a fluid can be defined as a material that can not remain motionless under the action offorces that leave its volume unchanged but otherwise act to deform it, as in the exampleabove.

Some materials can act as elastic solids when they are forced on short time scales, i.e.when they are sharply struck but deform in a manner like a liquid when the force operates

θ

Chapter 1 5

over a long time, like silly putty. Or, a less “silly” example is the mantle of the Earth whichsupports elastic (seismic) waves on short time scales (seconds) but deforms like a fluid onvery long time scales giving rise to mantle convection, sea-floor spreading and driftingcontinents. Air and water both act as fluids and one of the principal challenges for fluiddynamics is to obtain a useful mathematical representation of the relation between surfaceforces acting on pieces of fluid and the resulting deformations.

1.3 The continuum hypothesisOne of the pleasures of watching a moving fluid lies in the sinuous character of the

motion. The continuity of the water in a cascade, the slow rise of the smoke from a burningcigarette (please don’t smoke) that turns more complex but remains continuous as it risesand the rippling of the sea surface under a light wind are all examples of the tapestry-likepatterns of fluid motion. We intuitively think of the fluid involved as a continuum eventhough we are aware, intellectually, that the material is composed of atoms with large openspaces between them. Nevertheless, there are so many atoms on the scale of the motion thatis of interest to us, from fractions of a centimeter to thousands of kilometers that an averageover scales large compared to the inter atomic distances but small compared to themacroscopic scales of interest, allows us to establish a clear continuum with properties thatvary continuously on the scales we use to measure. So, for example, if we consider aproperty like the temperature, T, we can define an averaging scale l, that is much larger thaninter atomic or molecular distances but still small compared to the scale of our measuringinstruments.

Figure 1.3.1 The scale L is typical of our macroscopic measuring scale. The scale l isa scale used to produce a stable average of fluid properties like the temperature, T.

L

l

Chapter 1 6

Batchelor points out that even on scales for l of order 10-3 cm, and a volume 10-9 cm3,such a tiny volume would contain on the order of 3 101 0 air molecules in the atmosphereand this is more than enough to establish a thermodynamically stable definition oftemperature or any other such property.

This allows us to consider fields like the temperature field as continuous functions ofspace and time, e.g. T=T(x, t) {here we introduce the bold face notation forvectors—sometimes we will use arrows over the variable instead}. Further, we will usuallyassume that such functions are differentiable to whatever order we feel is necessary. If thereare discontinuities in our description of the fluid we consider them to be idealizations andsimplifications of rapid variations in the continuum and such variations are to be thought ofas occurring on scales much less than L but still large compared to the molecular scale.

An important consequence of this approach is that although it is a great simplificationto ignore the atomic nature of matter and treat the fluid as a continuous medium, certainproperties of the fluid like its viscosity, or thermal conductivity can not be predicted fromwithin the theory but must be specified as given properties of the fluid. And, they can becomplicated thermodynamic functions of temperature and pressure and that behavior must,as well, be specified external to the theory.

1.4 The fluid element.We will often speak of a fluid element or particle. What we will mean is an arbitrary

and arbitrarily small piece of the continuum that is a tagged piece of that fluid. Its volume isso small that its properties are uniform and it moves under the influence of the surroundingfluid. In principle, we mentally isolate it from its surroundings. Most important, we usuallyattribute to it a fixed mass.

Figure 1.4.1 The gray blob is a fluid element of fixed mass that deforms as it moves

A fluidelement

Chapter 1 7

We introduce this notion primarily because Newton’s Law of motion is most easilywritten in terms of force balances on a “particle” of fixed mass although, as we shall seethis is not necessary. It does allow us to consider the nature of many phenomena in anintuitive way if we can picture the dynamics of an isolated element. Like all simplifications ithas to be used with care.

1.5 KinematicsBefore discussing the dynamics of fluids we must first consider and agree how to

describe the fluid’s motion. This is the subject of kinematics. For a small solid object, like astone, the description is usually quite simple. We just describe the trajectory, say of thecenter of mass as a function X =X(t) in three dimensions and relate the forces to thevelocity and acceleration determined from our knowledge of X(t). In dealing with a fluid itis clearly not so simple. To paraphrase Gertrude Stein, there is no there there. We areclearly not interested in the center of gravity of the ocean, or not only in that, but in adescription of the whole moving continuum, an infinitude of fluid elements. We have foundtwo methods of description that are useful and illuminating. One is called the Eulerianmethod, the other is the Lagrangian method although both are probably due to Euler andlater exploited by Lagrange.

1.5.a The Eulerian approachThe fluid extends over some three dimensional space and exists in time. Consider any

property, P, for example the temperature. If x is the position vector with components {xj :

j= 1,2,3)♦ then we can describe P as P(xi,t). This gives us P at each location and at each time

without specifying which fluid element occupies that position at that time. In this descriptionwe do not keep track of individual fluid elements. This is a field description of the fluid andthe absence of information about the identity of the occupier of a position may seem a looseway to proceed but it has many advantages, principle among them that it does not require usto keep track of the extra information about the individual fate of particular elements. Afixed current meter, or a fixed anemometer all measure in an Eulerian context.

1.5b The Lagrangian approachAs in particle mechanics, the Lagrangian approach keeps track of individual fluid

elements as they move and describes the property P as a function of which particle it refersto and at each time. In this description each particle is given a name, i.e. a label, and thisserves to identify the fluid particle at all subsequent times. A convenient label is the position ♦ the coordinates (x1,x2,x3) are equivalent to (x,y,z)

Chapter 1 8

xi that the particle had a some initial instant, t=tO. Suppose we call that position Xi so thatXi=xi at t=tO. Each fluid element then changes its position as it moves but its label Xremains fixed.

1.5c The fluid trajectory

A simple way to define the trajectory of a fluid element is to specifyxi = xi (t,Xj ), where

xi (0,Xj ) = Xi(1.5. 1 a,b)

This is the trajectory equation for a particular element. It tells us where, at time t, is the fluidelement that initially (i.e. tO has been taken to be zero) was at position X. For all t > 0 thereis an inverse to this relation,

Xi = Xi (t, x j ) (1.5.2)

which tells us which particle is at position xi at time t. So, we can equally well describe anyproperty P in terms of the Lagrangian variables, P =P (X,t) and using (1.5.1) and (1.5.2) wecan go back and forth between the descriptions. Floats and radiosondes are essentiallyLagrangian measuring tools.

In actual observational situations the description of any property field is usuallyincompletely sampled. There are only a finite number of floats that can be available, currentmeter arrays are sparsely distributed, etc, so it is often a challenge to go back and forth fromone kinematic description to another.

1.6 Rates of change.Consider a property P(xi, t) in the Eulerian description. Its rate of change with respect

to time is central to the description of the fluid’s dynamics. We must keep in mind thatthere are several natural rates of change and they are all different. For example, when youare traveling on the way to Woods Hole from Boston you might detect a change in theweather, e.g. the temperature as you go along. You might detect that change not because thetemperature is changing with time at each location but because the temperature is changingwith distance and you are traveling in that temperature gradient. We need to distinguishbetween these rates of change.1) A rate of change at a fixed point is:

the local time derivative = ∂P∂t

⎞⎠⎟ xi

(1.6.1)

Chapter 1 9

This yields the rate of change with time at a fixed point of the property P. We shall usuallysuppress the subscript xi as understood. You can think of this as the change in P as a seriesof fluid elements move through the point xi and so it is possible that this rate of change canbe different from zero even if the property P is constant for each fluid element along itstrajectory.2) In general consider the rate of change of P for a moving observer (perhaps in a

convertible on Route 3). The observer moves with a velocity

v = vi{ }

such that for the observerdxidt

= vi

It is important to keep in mind that vi is not the fluid velocity. Thus, following the observerthe total rate of change as seen by the observer is:

dPdt

=∂P∂t

+∂P∂xii=1

∑ dxidt

(1.6.2)

ASIDE:We now introduce the summation convention or sometimes it is called the Einsteinsummation convention. When we have a product sum as in (1.6.2) where an a dummy index(in this case i) is summed over we understand that the sum is implied over the indexwithout explicitly writing the summation sign. Thus, for example, the inner product ofvectors A and B can be written:

AiB = AiBi

i=1

3

∑ ≡ AiBi

The rate of change of P given by (1.6.2) is then

dPdt

=∂P∂t

+∂P∂x j

dx jdt

=∂P∂t

+ vi∇Pthe rate of change of P seen by the observerdue to the observers motion in the gradient of P

(1.6.3)

Chapter 1 10

For example, a child on a slide whose elevation above the ground is h(x) will change his/her

height off the ground at a rate dh / dt = u∂h ∂x < 0.

Figure 1.6.1 A child going down a slide reduces her/his height at a rate that is the productof the forward velocity and the slope of the slide.

However, of particular interest is the rate of change of a property when the observer is thefluid itself, i.e. , the rate of change of a fluid property following a fluid element, or simplythe rate of change of the property for the fluid element. In that case

dxidt

= ui (xi ,t) (1.6.4)

where ui are the 3 components of the fluid velocity. Thus, for the fluid, the rate of change ofa property P following each fluid element is:

dPdt

=∂P∂t

+ ui∂P∂xi

=∂P∂t

+ ui∇P (1.6.5)

This derivative is the rate of change of P for the fluid particle that is occupying the positionxi at time t. This Eulerian representation of the derivative is variously called the total

derivative or the material derivative and is also sometimes written as DPDt

to distinguish this

derivative from that following a different observers rate of change. We will generally usethe notation in (1.6.5). The total derivative consists of two parts. The first term on the righthand side of (1.6.5) is the local time derivative. The second term involving the motion of thefluid in the gradient of the property P is called the advective derivative and the totalderivative is the sum of the local and advective parts.

The total derivative is central to fluid mechanics and please note that we need to knowthe fluid velocity as well as the space and time behavior of P to calculated the materialderivative of P. If both P and ui are unknown the material derivative involves the product oftwo unknowns and the fundamental nonlinearity of fluid mechanics and all the phenomena,like turbulence, weather, eddies, chaos etc. follow from this simple fact.

u

h

Chapter 1 11

Suppose that the property P is conserved for each fluid element so that its totalderivative is zero. It follows that at each spatial location:

∂P∂t

= − ui∇P (1.6.6)

So, at a fixed location an observer would see a change in the property as a parade offluid elements with different P pass that point. Think of a file of people of different heights,arranged in order of increasing height to the right. If the column moves to the right shorterand shorter people will occupy a given location and an observer will see constantlydecreasing heights occupy the observation position even though each person maintainshis/her total height (no beheadings allowed!)

Figure 1.6.2 A file of people moving to the right. If the people are arranged with increasingheight a fixed observer with see the height of people increasing at a fixed point. The heightof each person is fixed.

Suppose the property P is given by the relation:

P = Po cosωt sin kx (1.6.7)

where ω is the frequency 2π /T where T is the period and k is the wavenumber 2π / λ

where λ is the wavelength. Suppose the velocity in the x direction is U. Then the total

derivative of P is

grad (h) > 0

u

Observer location

Chapter 1 12

dPdt

= −Poω sinωt sin kx∂P∂t(local )

+UkPo cosωt coskx

u i∇P(advective)

=Po2(Uk +ω )cos(kx +ωt) + (Uk −ω )cos(kx −ωt)[ ]

(1.6.7)

The ratio of the local part of the time derivative to the advective part is of the order

∂P∂tui∇P

= O ωUk

⎛⎝⎜

⎞⎠⎟=cU

(1.6.8)

where c is the phase speed of the wave so the ratio measures the phase speed of the wavewith respect to the speed of the fluid. If the ratio c/U is large, the advective derivative can beignored to lowest order and the system will be linear.In the Lagrangian framework the total derivative following a fluid particle is simply

dPdt

=∂P∂t

⎞⎠⎟ Xi

(1.6.9)

For simplicity suppose that U is a constant. Thendxdt

=U ⇒ x =Ut + X (1.6.10)

so that P in (1.6.7) can be written, with a few algebraic identities and using (1.6.10)

P =Po2sin (kU +ω )t + X{ } + sin (kU −ω{ }t + X[ ] (1.6.11)

so that the total derivative (1.6.9) yields,

dPdt Lagrangian

=Po2

Uk +ω( )cos{(kU +ω )t + X} + (Uk −ω )cos{(kU −ω )t + X⎡⎣ ⎤⎦ (1.6.12)

It is easy to see that the transformation (1.6.10) shows the equivalence of (1.6.7) and(1.6.12).

Chapter 1 13

1.7 Streamlines

We define a streamline as a line in the fluid that is, at each instant, parallel to thevelocity. If d

x is a vector element along the streamline, this implies

dx = λ u (1.7.1)

where λ is an arbitrary constant. In component form it follows that,

dxu

=dyv

=dzw

= λ (1.7.2)

which can be thought of as two differential equations,

dzdx

=wu, dy

dx=vu

((1.7.3)

which with initial conditions yields a curve in space. This is the streamline.

Figure 1.7.1 A streamline. At every point of the streamline the velocity vector is tangent tothe line. (x,y,z) are the Cartesian coordinates and (u, v, w) are the corresponding velocitycomponents.

An equivalent representation can be used in which the arbitrary scalar λ is replaced by

the differential parameter ds so that the condition (1.7.1) is written as 3 ordinary differentialequations ,

x, u

y, v

z, w u

Chapter 1 14

dxds

= u,

dyds

= v,

dzds

= w.

(1.7.4 a,b,c)

and so avoiding the apparent singularity that occurs where u is zero. s is simply aparameter along the streamline. Since the streamline is constructed at an arbitrary fixedtime (and so will change its geometry if the velocity field is time dependent) it is frozen inplace, like an imaginary highway for the flow, and gives the direction the fluid element on itwould move if the velocity field were steady. So, we could interpret s as the time-likevariable for the motion of the fluid along the streamline if the flow were steady. Of course, ifthe flow is not steady the streamlines constructed this way give a clear picture of thedirection of the flow at this instant but do not give the trajectory or path of the fluid elementon it. The highway, so to speak, can keep changing direction with time and so the path of theparticle will diverge from the streamline.

For example, consider the two dimensional flow (w=0) for which,

u =Uo,

v = Vo cosk(x − ct)(1.7.5 a, b)

and for simplicity, Uo,Vo are constants. To find the fluid element trajectory we need tointegrate the Lagrangian trajectory equations,

dxdt

=Uo,⇒ x =Uot + X

⇒dydt

= Vo cos[k(Uo − c)t + kX],

or

y = Vo / kUo − c

sin k(x − ct) − sin kX{ }.

(1.7.6 a,b,c)

Since t is related to x by (1.7.6 a) the path of the fluid element is:

Chapter 1 15

y = Vo / kUo − c

sin k Uo − c{ } (x − X)Uo

+ kX⎡

⎣⎢

⎤

⎦⎥ − sin kX

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪(1.7.7)

On the other hand the stream line passing through the point x=X, y=0 at t=0, is determinedfrom the streamline equation,

dydx

=VoUo

cosk(x − ct),

⇒ y = Vo / kUo

{sin(k[x − ct]) − sin kX}

(1.7.8 a,b)

Figure 1.7.2 The trajectory (solid line) and the streamline in the x-y plane for the velocityfield given by (1.7.5)

Figure (1.7.2) shows the trajectory for the particle that a t=0 is at x=X=0, y=0 as the solid

line for Vo=1, Uo=2, and c =1 for k=π. The streamline at t=0 is shown passing through the

same initial point. Note that the streamline , while giving the direction of the velocity allalong the x axis at t=0 is not the subsequent path of the fluid element . The wavelength ofthe trajectory and its amplitude are quite different. An extreme case is shown in Figure 1.7.3when Uo-c is very small.

Chapter 1 16

Figure 1.7.3 As in the previous figure except that now c=1.95

The departure of the trajectory from the streamline is dramatic. As c approaches Uo the fluidelement, which moves in x with velocity Uo, remains fixed to the same phase of the wave,which moves with c, and so always sees the same value of v. This leads to an increasingdivergence of the trajectory from the streamline. In fact, and it is left to you as an exercise,in the limit as c Uo the trajectory is just a straight line whose slope, y/x, is just Vo/Uo. Thishas implications for the motion of individual air particles or water particles in the generallytime dependent, turbulent atmosphere and oceans. Even very orderly streamlines, usuallycoincident with isobars, can mask significantly more extreme particle transports overgreater distances than is evident from the streamline pattern.

Of course, if the flow is steady or nearly steady, the streamlines are a good image ofthe trajectories. Even when the flow is unsteady the streamlines accurately portray thedirection of the motion at that instant over the whole flow field.

Chapter 1 17

It is possible to construct a solution of (1.7.2) that consists of a surface composed of

streamlines. Let’s call that surface ψ(x,y,z) =C (suppressing the dependence on t which is

understood).

Figure 1.7.4 A portion of the surface ψ=constant showing the streamlines (the

stripes), one streamline extending beyond the surface and the direction normal to thesurface.

By definition of the surface,

ui∇ψ = 0. (1.7.9)

In fact, this is consistent with (1.7.1) and the condition that

∂ψ∂x

dx + ∂ψ∂y

dy + ∂ψ∂z

dz = dψ = 0 (1.7.10)

Note that if the flow is steady and so that the streamlines are also trajectories, (1.7.9) would

be equivalent to the statement that the property ψ is conserved for each fluid element. Now,

if we can construct one such surface composed of streamlines we can construct others.

Consider a surface φ= D which is independent of ψ so that

ui∇φ = 0. (1.7.11)

The two surfaces intersect in a line that is necessarily a streamline since both (1.7.9) and(1.7.11) must be simultaneously satisfied as shown in Figure 1.7.5.

ψ=C

u∇ψ

Chapter 1 18

Figure 1.7.5 The intersection of the two surfaces ψ and φ. The intersection line, by

construction is a streamline.

Since the velocity vector is perpendicular to both ∇ψ and ∇φ it follows that thevelocity at any location can be represented as,

u = γ∇φ × ∇ψ (1.7.12)

where, for now, γ is an arbitrary scalar function. At first sight this does not seem to be much

of an advance since we have replaced as unknowns the three scalar components of velocity

for the three unknown scalars γ,ψ and φ. But when we combine (1.7.12) with the statement

of mass conservation , which follows in the next chapter, this representation becomes muchmore useful and when the motion is in two dimensions this representation reducesimmediately to the common streamfunction you may have met before. If the motion is two

dimensional, i.e. if say, w=0, then φ can be chosen to be z+zo , for zo arbitrary, in which case

(1.7.12) becomes,

u = γ k̂ × ∇ψ (1.7.13)

where k̂ is a unit vector in the z direction.

ψ =const.

φ=const.

u

Chapter 1 19

1.8 The stream tube, streak lines

Consider a cylinder composed of all streamlines passing through a closed curve inspace, as shown in Figure 1.8.1. The resulting surface is a tube whose walls are composedof streamlines. This is a stream tube and by definition no flow exits the tube across itswalls.

Figure 1.8.1 A stream tube composed of streamlines that form the cylindrical wall of thetube. By construction there is no component of fluid velocity that cross that wall.

We have defined fluid trajectories and fluid streamlines and we have seen that whenthe flow is unsteady they are not the same. There is a third kinematic concept that is oftenused, the streakline. This is the line traced out in time by all the fluid particle s that pass aparticular point as a time goes on. It is left as an exercise for you to work out the streaklinefor the flow (1.7.5 a, b) that passes goes through the point x=0, y=0. The results can bequite surprising. For example, figure 1.8.2 shows the streamlines, trajectories and thestreaklines for fluid emanating from x=0,y=0.

The streakline would be the line of dye released at the point (0,0) as observed at somelater time downstream. From the figure it appears as if some kind of instability hasoccurred but looking at the generating velocity (1.7.5) that is clearly not the case. Can youexplain it? It is a good test of your understanding of the kinematics we have discussed sofar. Note that all the curves in figure (1.8.2) would coincide if c were zero.

These are streamlines

u

Chapter 1 20

Figure 1.8.2 The streamline (dashed), the trajectory (solid) and the streakline (dot-dashed) emanating from (x,y) = (0,0).